Osculating Invariant Circles

• Osculating invariant circles are the unique circles with second-order contact to a curve, providing local curvature approximations and forming nested, disjoint families.
• They are central in conformal and Lorentzian geometry, where they yield invariants like conformal arclength, curvature, and torsion.
• In symplectic dynamics, these circles model KAM phenomena and the accumulation of invariant circles along non-split separatrices under perturbative conditions.

Osculating invariant circles arise in several precise senses in the cited literature. For a smooth curve, the osculating circle at a point is the unique circle that has second-order contact with the curve there; for plane curves with strictly monotonic curvature, the Tait–Kneser theorem states that these circles are pairwise disjoint and nested (Ghys et al., 2012). In conformal geometry, osculating circles and spheres furnish Möbius-invariant local data and lead to the conformal arclength, conformal curvature, and conformal torsion (Dorn, 2023). In symplectic dynamics, by contrast, an invariant circle is a KAM curve, and such circles may accumulate, or osculate, along a non-split separatrix under perturbative Hamiltonian hypotheses (Katok et al., 2021). The topic therefore combines local contact geometry, Lorentzian models of circle spaces, conformal invariants, and dynamical accumulation phenomena.

1. Basic definitions and local geometry

Let $\gamma:I\to\mathbb R^2$ be a smooth, regular plane curve parameterized by arclength $s\in I$. Denote by $T(s)=\gamma'(s)$ the unit tangent, and by $N(s)$ the unit normal chosen so that $(T,N)$ is positively oriented. Its curvature is

$$\kappa(s)=\|\gamma''(s)\|, \qquad \gamma''(s)=\kappa(s)\,N(s),$$

and its radius of curvature is

$R(s)=1/\kappa(s),$

with $\kappa(s)>0$ on the arc under consideration. The center of curvature is

$z(s)=\gamma(s)+R(s)\,N(s),$

and the locus $z(s)$ is the evolute of $s\in I$0. The osculating circle at $s\in I$1 is the circle of radius $s\in I$2 centered at $s\in I$3; equivalently, it is the unique circle that has second-order tangency with $s\in I$4 at $s\in I$5 (Ghys et al., 2012).

For a smooth space curve $s\in I$6 in $s\in I$7, parametrized by arclength, the Frenet–Serret apparatus is

$s\in I$8

At each point $s\in I$9 the osculating circle $T(s)=\gamma'(s)$0 has radius $T(s)=\gamma'(s)$1 and center

$T(s)=\gamma'(s)$2

Its geometric significance is that $T(s)=\gamma'(s)$3 is the unique circle through $T(s)=\gamma'(s)$4 that has second-order contact with the curve; it is the best planar circular approximation to the curve near $T(s)=\gamma'(s)$5, its curvature equals that of the space curve, and its plane is the osculating plane $T(s)=\gamma'(s)$6 (Dorn, 2023).

The monotonic curvature condition for a plane curve means that $T(s)=\gamma'(s)$7 is either everywhere positive or everywhere negative on $T(s)=\gamma'(s)$8, with $T(s)=\gamma'(s)$9. This hypothesis is the classical setting in which disjointness and nesting of osculating circles become rigid.

2. Tait–Kneser theorem and nested osculating disks

A standard formulation of the Tait–Kneser theorem is as follows. Let $N(s)$0 be a $N(s)$1-smooth regular curve with strictly monotonic positive curvature $N(s)$2 for $N(s)$3. Denote by $N(s)$4 the osculating circle of radius $N(s)$5 centered at $N(s)$6. Then for any $N(s)$7 one has

$N(s)$8

In particular the disks

$N(s)$9

are strictly nested and pairwise disjoint: if $(T,N)$0 then $(T,N)$1 (Ghys et al., 2012).

The proof proceeds through the evolute. On an arc with monotonic curvature the evolute $(T,N)$2 has no cusps and is a regular curve. One has

$(T,N)$3

where $(T,N)$4 is the unit tangent to the evolute. Hence the arclength of the evolute between $(T,N)$5 and $(T,N)$6 is

$(T,N)$7

Since on any smooth convex arc the chord length is strictly less than the arclength,

$(T,N)$8

Because $(T,N)$9 is monotonic, $$\kappa(s)=\|\gamma''(s)\|, \qquad \gamma''(s)=\kappa(s)\,N(s),$$0 does not change sign, so the sign of $$\kappa(s)=\|\gamma''(s)\|, \qquad \gamma''(s)=\kappa(s)\,N(s),$$1 is known. Equivalently, if $$\kappa(s)=\|\gamma''(s)\|, \qquad \gamma''(s)=\kappa(s)\,N(s),$$2, then

$$\kappa(s)=\|\gamma''(s)\|, \qquad \gamma''(s)=\kappa(s)\,N(s),$$3

which is exactly the condition that the closed disk of radius $$\kappa(s)=\|\gamma''(s)\|, \qquad \gamma''(s)=\kappa(s)\,N(s),$$4 about $$\kappa(s)=\|\gamma''(s)\|, \qquad \gamma''(s)=\kappa(s)\,N(s),$$5 lies strictly inside the disk of radius $$\kappa(s)=\|\gamma''(s)\|, \qquad \gamma''(s)=\kappa(s)\,N(s),$$6 about $$\kappa(s)=\|\gamma''(s)\|, \qquad \gamma''(s)=\kappa(s)\,N(s),$$7 (Ghys et al., 2012).

The same source emphasizes an invariant-foliation phenomenon. The union of osculating circles for $$\kappa(s)=\|\gamma''(s)\|, \qquad \gamma''(s)=\kappa(s)\,N(s),$$8 fills an annulus in the plane whose leaves are these circles, and the curve $$\kappa(s)=\|\gamma''(s)\|, \qquad \gamma''(s)=\kappa(s)\,N(s),$$9 passes from circle to circle, always tangent to them. However, this foliation is not given by a global $R(s)=1/\kappa(s),$0-function: any differentiable function $R(s)=1/\kappa(s),$1 in the annulus which is constant on each osculating circle must be constant. Illustrative examples include a logarithmic spiral, which has strictly monotonic curvature and displays a one-parameter family of nested osculating circles, and an ellipse, whose evolute has four cusps at its vertices, showing that monotonicity of $R(s)=1/\kappa(s),$2 breaks precisely at extrema of curvature (Ghys et al., 2012).

3. Lorentzian formulations and 3-parameter variations

A compact reformulation identifies an oriented Euclidean circle of signed radius $R(s)=1/\kappa(s),$3 and center $R(s)=1/\kappa(s),$4 with the point

$R(s)=1/\kappa(s),$5

where $R(s)=1/\kappa(s),$6 is endowed with the Lorentzian metric of signature $R(s)=1/\kappa(s),$7

$R(s)=1/\kappa(s),$8

or equivalently with inner product

$R(s)=1/\kappa(s),$9

Two circles represented by $\kappa(s)>0$0 and $\kappa(s)>0$1 are nested if and only if

$\kappa(s)>0$2

with equality precisely when they are tangent (Bor et al., 2021).

If $\kappa(s)>0$3 is a smooth regular curve parametrized by arc-length $\kappa(s)>0$4, with curvature $\kappa(s)>0$5 everywhere and $\kappa(s)>0$6 of one sign, then the osculating circle has radius $\kappa(s)>0$7 and center

$\kappa(s)>0$8

The associated curve of osculating circles is

$\kappa(s)>0$9

A direct computation gives

$z(s)=\gamma(s)+R(s)\,N(s),$0

and since $z(s)=\gamma(s)+R(s)\,N(s),$1, the vector $z(s)=\gamma(s)+R(s)\,N(s),$2 lies on the light-cone of $z(s)=\gamma(s)+R(s)\,N(s),$3. Hence

$z(s)=\gamma(s)+R(s)\,N(s),$4

Strict monotonicity of $z(s)=\gamma(s)+R(s)\,N(s),$5 implies $z(s)=\gamma(s)+R(s)\,N(s),$6, so $z(s)=\gamma(s)+R(s)\,N(s),$7 is a regular null curve. A standard Lorentz-geometry fact then implies that the squared Lorentz-distance between any two points of $z(s)=\gamma(s)+R(s)\,N(s),$8 is nonnegative, and is strictly positive unless the curve between them is itself a straight null-line segment. In the osculating-circle setting one obtains

$z(s)=\gamma(s)+R(s)\,N(s),$9

which is exactly the nesting criterion for the corresponding circles (Bor et al., 2021).

The same Lorentzian outline extends to two further 3-parameter families. In centroaffine geometry, the analogues of circles are the central conics

$z(s)$0

with Lorentzian metric

$z(s)$1

Two central conics of the same type are nested if and only if

$z(s)$2

with equality at tangency. The osculating Hooke conic of a star-shaped plane curve with nonzero monotone centro-affine curvature again gives a regular null curve, and the same chord-inequality yields pairwise disjoint and nested osculating Hooke conics. For Kepler conics, parametrized by planes

$z(s)$3

intersecting the light-cone $z(s)$4 in $z(s)$5, the metric is again

$z(s)$6

and two Kepler conics are disjoint, equivalently nested, if and only if the Lorentzian norm of their difference is positive. Their osculating family also forms a regular null curve, so the same method proves a Kepler analogue of Tait–Kneser (Bor et al., 2021).

4. Conformal invariants from osculating circles and spheres

Möbius transformations in $z(s)$7 map circles and spheres into circles or spheres and preserve the order of contact. In particular, the osculating circle at $z(s)$8 is carried into the osculating circle of the image curve at $z(s)$9. By the kinematics of the family of osculating circles $s\in I$00 one may extract differential invariants invariant under the full 10-parameter conformal group of $s\in I$01 (Dorn, 2023).

For a generic smooth curve, define

$s\in I$02

which never vanishes for a generic smooth curve. Then: $s\in I$03 defines the conformal arclength;

$s\in I$04

defines the conformal curvature; and

$s\in I$05

defines the conformal torsion. All three functions $s\in I$06, $s\in I$07, and $s\in I$08 are invariant under any conformal transformation of $s\in I$09 (Dorn, 2023).

At a cusp of a piecewise-smooth curve, meaning two smooth legs meeting with a nonzero opening angle, one has on each leg an osculating circle $s\in I$10, $s\in I$11 and similarly osculating spheres $s\in I$12, $s\in I$13. One may form four pairwise conformal invariants. For the two circles,

$s\in I$14

For the two spheres,

$s\in I$15

For the mixed circle–sphere pairs,

$s\in I$16

A direct analysis shows that among the four $s\in I$17 there are exactly three independent conformal parameters attached to the cusp. One convenient choice is

$s\in I$18

These three numbers are invariant under any global conformal map, and they reduce to the usual cusp-angle $s\in I$19 in the planar case (Dorn, 2023).

A concrete example is given by two planar circular arcs of the same radius $s\in I$20, one lying in the $s\in I$21-plane and the other in the $s\in I$22-plane, joined at the point $s\in I$23. Each arc has constant $s\in I$24 and $s\in I$25, so along each leg the conformal invariants $s\in I$26 vanish or are ill-defined; they are conformal vertices. The tangent vectors satisfy $s\in I$27 and $s\in I$28, so $s\in I$29. The corresponding sphere–sphere invariant is

$s\in I$30

which remains unchanged under a Möbius transformation such as inversion in a sphere about the origin (Dorn, 2023).

5. The space of circles and Möbius-invariant moving frames

A second conformal description realizes the conformal 3-sphere $s\in I$31 as the set of null rays in Minkowski space

$s\in I$32

with Lorentzian form

$s\in I$33

In an affine chart determined by a fixed light-like $s\in I$34, a Euclidean curve $s\in I$35 lifts to a null curve in $s\in I$36 with

$s\in I$37

The osculating circle at $s\in I$38 is then the intersection of the light cone with the time-like 3-plane

$s\in I$39

or dually the light-like 2-plane $s\in I$40 (Langevin et al., 2011).

Circles in $s\in I$41 are in one-to-one correspondence with oriented space-like 2-planes in $s\in I$42. In Plücker coordinates such a plane is represented by a pure 2-vector

$s\in I$43

satisfying

$s\in I$44

The second equation defines the quadric $s\in I$45, the space of circles. If a curve is vertex-free, its osculating spheres

$s\in I$46

trace a space-like curve, and one obtains the osculating-circle curve

$s\in I$47

Moreover,

$s\in I$48

so $s\in I$49 is a null-curve in the circle-quadric (Langevin et al., 2011).

The Möbius-invariant moving frame is organized by the conformal arc-length $s\in I$50, determined up to $s\in I$51 by

$s\in I$52

and by the arclength $s\in I$53 of the osculating-sphere curve. The conformal torsion is

$s\in I$54

and one also has

$s\in I$55

where $s\in I$56 is the conformal curvature. With the isotropic orthonormal frame

$s\in I$57

defined by

$s\in I$58

the Frenet system has matrix

$s\in I$59

This expresses exactly how $s\in I$60 and $s\in I$61 enter as the two non-trivial Möbius-invariant functions (Langevin et al., 2011).

The same framework yields a normal form for a generic curve and a characterization of canal surfaces. In particular, a canal surface is an envelope of a one-parameter family of spheres $s\in I$62, its characteristic circles are the intersections of consecutive spheres, and conversely any curve $s\in I$63 whose derivative still satisfies the purity relations arises from a unique canal surface (Langevin et al., 2011).

6. Invariant circles and separatrix accumulation in symplectic dynamics

A distinct use of invariant circles occurs in smooth symplectic dynamics. Let $s\in I$64 be a $s\in I$65, symplectic diffeomorphism with a hyperbolic fixed point $s\in I$66, meaning that $s\in I$67 has two real eigenvalues $s\in I$68 and $s\in I$69. Its stable and unstable manifolds are

$s\in I$70

A non-split separatrix $s\in I$71 associated to $s\in I$72 is a compact, connected, $s\in I$73-invariant set homeomorphic to $s\in I$74 such that

$s\in I$75

is a single connected 1-manifold; non-split means exactly that the stable and unstable branches coincide along $s\in I$76 (Katok et al., 2021).

In the standard open annulus

$s\in I$77

an invariant circle or KAM curve for a symplectic diffeomorphism $s\in I$78 is a non-self-intersecting, $s\in I$79 embedded circle

$s\in I$80

such that $s\in I$81 and the induced map on $s\in I$82 is $s\in I$83-conjugate to an irrational rotation. If the rotation number satisfies a Diophantine estimate, one calls $s\in I$84 a KAM circle. A family $s\in I$85 accumulates, or osculates, along $s\in I$86 if for every $s\in I$87 there are circles with $s\in I$88 small enough and

$s\in I$89

where $s\in I$90 is the Hausdorff distance (Katok et al., 2021).

The perturbative accumulation theorem is stated for an autonomous Hamiltonian vector field

$s\in I$91

with

$s\in I$92

so that its time-1 map has a hyperbolic fixed point $s\in I$93 and a non-split separatrix $s\in I$94. Let

$s\in I$95

be a smooth time-periodic Hamiltonian perturbation tangent to $s\in I$96, and define

$s\in I$97

Assume the twist condition

$s\in I$98

Then for each integer $s\in I$99 there exists $T(s)=\gamma'(s)$00 so that for all $T(s)=\gamma'(s)$01, $T(s)=\gamma'(s)$02 admits a Cantor family of $T(s)=\gamma'(s)$03-invariant KAM circles, the set of Diophantine rotation numbers has positive Lebesgue measure, these circles accumulate $T(s)=\gamma'(s)$04, and in any neighborhood $T(s)=\gamma'(s)$05 of $T(s)=\gamma'(s)$06 their union covers a set of positive two-dimensional Lebesgue measure in $T(s)=\gamma'(s)$07 (Katok et al., 2021).

The proof uses a Birkhoff–Sternberg normal form near the hyperbolic fixed point, a symplectic Sternberg linearization, a fundamental domain and return map, and log-coordinates

$T(s)=\gamma'(s)$08

in which the return map is, after smooth coordinate change and suitable renormalization, a $T(s)=\gamma'(s)$09-small perturbation of an integrable twist map

$T(s)=\gamma'(s)$10

with $T(s)=\gamma'(s)$11 bounded away from zero. Rüssmann–Moser then yields invariant graphs, and tracing back the coordinate changes gives

$T(s)=\gamma'(s)$12

The same work also constructs a $T(s)=\gamma'(s)$13 symplectic diffeomorphism with a Lyapunov unstable non-split separatrix for which no invariant circles accumulate on $T(s)=\gamma'(s)$14, showing that the perturbative smallness assumption is essential (Katok et al., 2021).

7. Degeneracies, special points, and the scope of nesting theorems

The literature places Tait–Kneser type nesting results in duality with least-number theorems for special points. For closed strictly convex plane curves, the 4-vertex theorem states that there are at least four curvature extrema. For conics, the 6-sextactic theorem asserts that a closed convex oval has at least six sextactic points. For diffeomorphisms of $T(s)=\gamma'(s)$15, the 4-Schwarzian theorem says that any diffeomorphism has at least four zeros of its Schwarzian derivative. For cubics, one has a 10-extactic-point theorem near a cubic oval. These theorems are described as expressing a duality between nesting or disjointness results on arcs free of degeneracies, and least-number special-point results on closed curves (Ghys et al., 2012).

The underlying osculating families vary with the geometric setting. The 5-parameter family of nondegenerate projective conics can osculate a plane curve to order four, and a point is sextactic if the osculating conic has order at least five contact. On an arc free of sextactic points two nearby osculating conics intersect only at the osculation point, with total multiplicity four, and have no other real intersections; hence their interiors in $T(s)=\gamma'(s)$16 are nested and disjoint. The 3-parameter group $T(s)=\gamma'(s)$17 of fractional-linear transformations osculates a diffeomorphism $T(s)=\gamma'(s)$18 to second order at each point, while hyperosculation to third order occurs exactly at zeros of the Schwarzian derivative $T(s)=\gamma'(s)$19; if $T(s)=\gamma'(s)$20 does not vanish on an arc, the graphs of the corresponding osculating Möbius transforms are pairwise disjoint. For cubic curves, if a smooth $T(s)=\gamma'(s)$21 has no extactic points, then consecutive osculating ovals meet only at the osculation point, counted with multiplicity ten, and nowhere else, hence are nested and disjoint (Ghys et al., 2012).

A closely related result appears for Kepler conics. If $T(s)=\gamma'(s)$22 is a simple closed star-shaped curve, then a vertex is a point where the osculating Kepler conic has third-order contact. In polar coordinates $T(s)=\gamma'(s)$23, with $T(s)=\gamma'(s)$24, every Kepler conic satisfies

$T(s)=\gamma'(s)$25

so the vertices are exactly the zeros of the smooth $T(s)=\gamma'(s)$26-periodic function $T(s)=\gamma'(s)$27. By the Sturm–Hurwitz argument one deduces that there are at least four distinct vertices (Bor et al., 2021).

Taken together, these results show a broad pattern. On arcs where the relevant degeneracies are absent—vertices, sextactic points, zeros of the Schwarzian derivative, or extactic points—the osculating objects form disjoint and nested families. On closed curves or global periodic objects, the same geometry forces the existence of special points where monotonicity or generic contact fails. A plausible implication is that “osculating invariant circles” is best understood not as a single rigid notion, but as a family of constructions in which local second-order contact, Lorentzian or Möbius invariance, and global nesting or accumulation are linked by the geometry of the corresponding parameter space.